MATH 2924 - Differential and Integral Calculus II (Honors), Sec. 040 & 041
- Fall 2015
MWF 1:30-2:20 in 809 PHSC (Lectures);
TR 3-3:50 in 117 PHSC (Discussions)
Course instructor: Prof. Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.
Discussion section instructor: Devin W. Mitchell, 1028 PHSC, dmitchell AT math.ou.edu
Office Hours:
M 2:30-3:30, W 11:20-12:20, or by appointment, in 802 PHSC (N.P.);
T 2-3 p.m., Fri 11-12, in 1028 PHSC (D.M.)
First day handout
OU Math Center (PHSC 209) - open MWR 9:30-5:30, T 9:30-7, F 9:30-3:30, S 3-7
Course catalog description:
Prerequisite: 1914 with a grade of C or better. Duplicates two hours of 2423
and two hours of 2433. Further applications of integration,
the natural logarithmic and exponential functions, indeterminate forms,
techniques of integration, improper integrals, parametric curves
and polar coordinates, infinite sequences and series. (F, Sp, Su)
Text:
J. Stewart, Calculus (7th ed), Brooks/Cole, 2012, ISBN 978-0-8400-5818-8
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Course content:
-
Inverse functions:
inverse functions, natural logarithm and natural exponential functions,
general logarithm and general exponential functions,
exponential growth and decay, inverse trigonometric functions,
hyperbolic functions, l'Hospital's rule.
-
Techniques of integration:
integration by parts, trigonometric integrals, trigonometric substitutions,
partial fractions, improper integrals, approximate integration, arc length.
-
Infinite sequences and series:
sequences, series, integral test, comparison test,
alternating series, absolute convergence (root and ratio tests),
power series, representation of functions as power series, Taylor and Maclaurin series,
Taylor polynomials.
-
Parametric equations and polar coordinates:
curves defined by parametric equations, calculus with parametric curves,
polar coordinates, conic sections,
areas and lengths in polar coordinates.
-
Vectors and the geometry of space:
three-dimensional coordinate systems, vectors,
dot and cross products, applications.
Homework:
-
Homework 1 (problems given on August 24, 26, 28), due September 2 (Wednesday)
-
Homework 2 (problems given on August 31, September 2, 4), due September 9 (Wednesday)
-
Homework 3 (problems given on September 9, 11, 14), due September 18 (Friday)
-
Homework 4 (problems given on September 16, 18, 21), due September 23 (Wednesday)
-
Homework 5 (problems given on September 25, 28, 30), due October 5 (Monday)
-
Homework 6 (problems given on October 2, 5, 7), due October 14 (Wednesday)
-
Homework 7 (problems given on October 12, 14, 16), due on October 21 (Wednesday)
-
Homework 8 (problems given on October 19, 21, 23), due on October 28 (Wednesday)
-
Homework 9 (problems given on October 26, 28, November 2, 4), due on November 9 (Monday)
-
Homework 10 (problems given on November 6, 9, 11), due on November 16 (Monday)
-
Homework 11 (problems given on November 13, 16, 18), due on November 23 (Monday)
-
Homework 12 (problems given on November 20, 23, 30), due on December 2 (Wednesday)
-
Homework 13 (problems given on December 2, 4, 7, 9), due on December 11 (Friday)
Content of the lectures:
-
Lecture 1 (Mon, Aug 24):
Inverse functions:
domain and range of a function; one-to-one functions;
horizontal line test (whether a function is one-to-one);
inverse function of a one-to-one function; cancellation equations;
examples; derivatives of inverse functions [page 10 of Sec. 1.1; pages 384-387 of Sec. 6.1]
Homework:
Exercises 6.1 / 14, 18,
19,
20, 22
FFT:
Exercises
6.1 /
6,
15,
21,
31
-
Discussion 1a (Tue, Aug 25):
Inverse functions (cont.):
derivative of inverse functions, examples
[pages 388-389 of Sec. 6.1]
-
Lecture 2 (Wed, Aug 26):
The natural logarithmic function:
definition of the function ln x by an integral,
elementary properties of ln x,
laws of logarithms;
sketching the graph of ln x
(by proving that ln x is monotonically increasing
and finding its limits as x→0+ and as x→∞;
challenge: can you use some mathematical result to prove the existence
of a number such whose natural logarithm is 0
[pages 421-423 of Sec. 6.2*]
Homework:
Exercises 6.1 / 37, 45, 50;
6.2* / 16, 18, 20, 48, 50, 68, 74
FFT:
Exercises
6.1 /
24,
41,
43;
6.2* /
1, 6,
70,
71,
89
-
Discussion 2a (Thu, Aug 27):
The natural logarithmic function (cont.):
using the Intermediate Value Theorem to prove the existence of a number
e such that ln e=1; examples
[page 89 of Sec. 1.8, page 423 of Sec. 6.2*]
-
Lecture 3 (Fri, Aug 28):
The natural logarithmic function (cont.):
examples of using the laws of logarithms; proof that the derivative
of ln|x| is 1/x,
and that the indefinite integral of 1/x is ln|x|+C;
integral of cot x; logarithmic differentiation;
Newton's method for computing the root of an algebraic equation
[pages 424-427 of Sec. 6.2*; Sec. 3.8]
Homework:
Exercises
6.2* /
36, 42, 64, 73, 80, 81, 85 (n positive integer),
89
FFT:
Exercises
6.2* /
25,
32, 34, 53 (solve ƒ(x)=(x−4)2−ln x=0), 77;
give an explicit counterexample to the following statement:
if a function ƒ(x) has an inverse function ƒ−1(x),
then ƒ(x) is either increasing or decreasing.
The complete Homework 1
(problems given on August 24, 26, and 28) is due on September 2 (Wednesday).
-
Lecture 4 (Mon, Aug 31):
The natural logarithmic function (cont.):
computation of the value of ln 1.003 by approximate
computation of the value of the integral of 1/t
from 1 to 1.003 by using left and right Riemann sums
or by trapezoidal method; the true value of ln 1.003
is smaller than the value produced by the trapezoidal method
because the function 1/t is concave up on the interval [1,1.003].
The natural exponential function:
invertibility of the function ln (coming from the fact
that ln'(x)>0),
definition of the function exp as the inverse
of the function ln,
domain and range of the function exp;
properties of the function exp:
exp(x1+x2)=exp(x1)exp(x2),
exp(x1−x2)=exp(x1)/exp(x2),
exp(rx)=(exp x)r (for r rational);
definition of the number e
as the unique solution of ln(x)=1;
exp(x)=ex;
derivative and integral of exp; examples
[Sec 6.3*]
Homework:
Exercises
6.3* /
8,
12,
16, 22, 26, 42, 52,
54,
88, 92
FFT:
Exercises
6.3* /
19,
30,
37,
43,
85
-
Lecture 5 (Wed, Sep 2):
General logarithmic and exponential functions:
defining ax as
ex ln a
for any a>0; laws of exponents;
derivative of ax;
integral of ax;
derivation of the power rule using logarithmic differentiation;
derivatives of xx
and more expressions of the form
[ƒ(x)]g(x);
general logarighmic functions loga;
change of base formula;
derivative of logax
[pages 437-442 of Sec. 6.4*]
Homework:
Exercises
6.3* /
55,
60;
6.4* / 8, 18, 24, 30, 42, 50
FFT:
Exercises
6.3* / 63;
6.4* /
17,
37,
45,
49,
54
-
Lecture 6 (Fri, Sep 4):
General logarithmic and exponential functions (cont.):
the number e as a limit
[page 443 of Sec. 6.4*]
Historical digression:
Kepler's
laws of planetary motion
(1609, 1619);
Newton's
derivation of Kepler's laws based on the
law of universal gravitation
(1687);
discovery of Neptune
(1846) based on the calculations of
Le Verrier and
Adams
based on studies of the irregularities in the orbit of Uranus;
slide rule
- based on the formula
ln(xy)=ln x+ln y
Homework:
Exercises
6.4* / 48, 56;
Ch 6 Review (pages 481-484) / 34, 42, 68, 69, 106, 114, 121
FFT:
Exercises
6.4* / 51;
Ch 6 Review (pages 481-484) / 49-52, 113
The complete Homework 2
(problems given on August 31, September 2, and 4) is due on September 9 (Wednesday).
-
Lecture 7 (Wed, Sep 9):
Exponential growth and decay:
simplest ideas in modeling population dynamics
- the rate of change, dP/dt, is proportional
to the population P(t), so that
dP/dt=kP;
deficiency of this model - the population
P(t)=P(0)ekt
grows unboundedly as t→∞;
correcting the model - introducing a correction accounting for
the finite resources - logistic equation
dP/dt=kP(1−P/A),
where A is the carrying capacity of the system;
behavior of the solutions of the logistic equation:
P(t)→A as t→∞
for every initial population P(0)>0;
radioactive decay:
dm/dt=km where k=const<0;
half-life T1/2 of a radioactive isotope;
relating the half-life of a radioactive isotope with the decay constant k
[pages 446-448 of Sec. 6.5]
Inverse trigonometric functions:
redefining the function sin to make it one-to-one
by restricting its domain to [−π/2,π/2];
defining the function arcsin (or sin−1)
as the inverse function of sin:[−π/2,π/2]→[−1,1];
cancellation equations
arcsin(sin(x))=x for all x∈[−π/2,π/2],
sin(arcsin(x))=x for all x∈[−1,1];
proof that cos(arcsin(x))=(1−x2)1/2
for all x∈[−1,1];
derivation of the formula for the derivative of arcsin
[pages 453, 454 of Sec. 6.6]
Digression on the importance of conservation laws in physics:
the conservation of energy
as a consequence of the fact that the physical laws are invariant (i.e., do not change)
under a translation in time
(in a mathematical language, the conservation of energy is a
particular case
of the so-called
Noether theorem);
1930 conjecture of
Pauli
(see an
exerpt from his original letter)
that the observed violation of conservation laws in
beta decay
is due to an elementary particle that is difficult to detect experimentally;
the particle - called
neutrino
by Fermi
- was discovered in 1956 by
Cowan
and
Reines.
Homework:
Exercises
6.5 / 4, 11;
6.6 / 7, 12, 27, 28, 60, 61, 68
FFT:
Exercises
6.5 /
3,
9;
6.6 / 11,
25,
43, 65, 67
-
Lecture 8 (Fri, Sep 11):
Inverse trigonometric functions (cont.):
redefining the function cos to make it one-to-one
by restricting its domain to [0,π];
defining the function arccos (or cos−1)
as the inverse function of cos:[0,π]→[−1,1];
cancellation equations
arccos(cos(x))=x for all x∈[0,π],
cos(arccos(x))=x for all x∈[−1,1];
proof that sin(arccos(x))=(1−x2)1/2
for all x∈[−1,1];
derivation of the formula for the derivative of arccos;
redefining the function tan to make it one-to-one
by restricting its domain to (−π/2,π/2);
defining the function arctan (or tan−1)
as the inverse function of tan:(−π/2,π/2)→R;
cancellation equations
arctan(tan(x))=x for all x∈(−π/2,π/2),
tan(arctan(x))=x for all x∈R;
proof that cos(arctan(x))=(1+x2)1/2
for all x∈R;
derivation of the formula for the derivative of arctan
[pages 455-459 of Sec. 6.6]
Homework:
Exercises
6.6 / 9, 20, 34, 38, 46, 62, 64, 77
FFT:
Exercises
6.6 / 1, 5, 10,
13,
45,
63,
70, 71
-
Lecture 9 (Mon, Sep 14):
Hyperbolic functions:
definition; basic properties - parity (i.e., whether a function is even/odd), values at 0,
limits at ∞ and −∞;
basic relation:
cosh2x−sinh2x=1;
expressions for sinh(x+y) and cosh(x+y)
in terms of sinh(x), sinh(y), cosh(x), and cosh(y);
derivatives of hyperbolic functions;
inverse hyperbolic functions;
expressing sinh−1(x) in terms of logarhtm and square root functions
[pages 462-465 of Sec. 6.7]
Homework:
Exercises
6.7 / 18, 19, 20, 32, 38, 49, 52,
55,
62, 63
FFT:
Exercises
6.7 / 3,
9,
17,
23, 28,
35
The complete Homework 3
(problems given on September 9, 11, and 14) is due on September 18 (Friday).
-
Lecture 10 (Wed, Sep 16):
Hyperbolic functions (cont.):
deriving the formula for the derivative of sinh−1x
in two ways: (1) by using the fact it is the inverse function of sinh(x)
and that d(sinh x)/dx=cosh(x),
and (2) by using the expression for sinh−1x
in terms of logarithm;
examples
[pages 465-467 of Sec. 6.7]
Indeterminate forms and l'Hospital's rule:
indeterminate forms:
0/0, ∞/∞, 0·∞, ∞−∞,
00, ∞0, 1∞
[pages 469, 474 of Sec. 6.8]
Homework:
Exercises
6.7 / 40, 42, 59, 65, 66;
6.8 / 2, 4, 8, 18, 23 (in 8, 18, and 23 you are not allowed to use l'Hospital!)
FFT:
Exercises 6.7 / 25, 29(b);
6.8 / 1, 3
-
Lecture 11 (Fri, Sep 18):
Indeterminate forms and l'Hospital's rule (cont.):
examples of using l'Hospital's rule for different types of indeterminate forms
[pages 469-475 of Sec. 6.8]
Homework:
Exercises
6.8 / 22, 44, 51, 56, 62, 91, 94,
99
FFT:
Exercises
6.8 /
20,
25,
45,
50,
57,
71;
Ch 6 Concept Check (pages 480-481);
Ch 6 True-False Quiz (page 481)
-
Lecture 12 (Mon, Sep 21):
Mathematical induction and other methods used in mathematical proofs:
method of mathematical induction; examples;
to prove that statement P implies statement Q,
prove the contrapositive, i.e., that (not Q) implies (not P);
reductio ad absurdum: to prove that statement P is correct,
prove that the statements (not P) leads to contradiction.
Homework:
Exercises
Ch 6 Review (pages 481-484) / 30, 38, 75, 96, 110, 112
FFT:
Exercises
Ch 6 Review (pages 481-484) / 16, 48, 59, 119, 120
The complete Homework 4
(problems given on September 16, 18, and 21) is due on September 23 (Wednesday).
-
Lecture 13 (Wed, Sep 23):
Mathematical induction:
idea of the method of mathematical induciton; examples
[pages A36, A37 of Appendix E]
Integration by parts:
derivation of the method of integration by parts from the product rule;
elementary examples
[pages 488-489 of Sec. 7.1]
-
Discussion 13a (Thu, Sep 24):
Exam 1
[on Sec. 6.1, 6.2*-6.4*, 6.5-6.8,
covered in Lectures 1-12]
-
Lecture 14 (Fri, Sep 25):
Integration by parts:
further examples; definite integrals;
deriving recursion relations by using integration by parts
[pages 489-492 of Sec. 7.1]
Homework:
Exercises
7.1 / 1,
15,
22,
26,
33,
39,
53,
additional problems (the additional problems
are not FFT problems, and should be turned in with the regular homework)
FFT:
Exercises
7.1 /
3,
12,
17,
20,
24,
38,
51,
55,
67
-
Lecture 15 (Mon, Sep 28):
Integration by parts (cont.):
solving integral of (1−x2)1/2
(1) by using integraiton by parts and
(2) by interpreting the integral of (1−t2)1/2
from 0 to x as the area between the unit circle, the t-axis,
the vertical axis, and the vertical line t=x.
Trigonometric integrals:
integrating products of sines and cosines;
deriving all trigonometric formulas from the identities
sin(x+y)=sin(x)cos(y)+cos(x)sin(y),
cos(x+y)=cos(x)cos(y)−sin(x)sin(y)
[pages 495-497 of Sec. 7.2]
Homework:
Exercises
7.1 / 42, 49,
70;
7.2 /
6,
11,
15, 16,
41,
55
FFT:
Exercises
7.1 /
61;
7.2 /
3,
7
-
Lecture 16 (Wed, Sep 30):
Trigonometric integrals (cont.):
integrating products of tangents and secants,
examples; integrating tan(x), sec(x),
tan3(x), sec3(x);
integrating products of sines and cosines
[pages 497-500 of Sec. 7.2]
Trigonometric substition:
integrating (1−x2)1/2
by using the substitution x=sin(θ);
integrating (1+x2)1/2
by using the substitution x=tan(θ)
or the substitution x=sinh(θ)
[pages 502-504 of Sec. 7.3]
Homework:
Exercises
7.2 /
21, 22,
24, 33, 46;
7.3 /
7,
11,
31(a),
31(b)
FFT:
Exercises
7.2 /
29,
48;
7.3 /
29, 39
The complete Homework 5
(problems given on September 25, 28, and 30) is due on October 5 (Monday).
-
Lecture 17 (Fri, Oct 2):
Trigonometric substition (cont.):
integrating (x2−1)1/2
by using the substitution x=sec(θ)
or the substitution x=cosh(θ)
[pages 505-507 of Sec. 7.3]
Homework:
Exercises
7.3 /
13,
14,
17
FFT:
Exercises
7.3 /
22,
23
-
Lecture 18 (Mon, Oct 5)
Integration of rational functions by partial fractions:
reducing a rational function P(x)/Q(x)
to its proper form,
S(x)+R(x)/Q(x),
where S, R, and Q are polynomials
with deg(R) smaller than deg(Q);
partial fraction expansion when the denominator is a product
of linear factors
[pages 508-512 of Sec. 7.4]
Homework:
Exercises
7.4 / 7,
11,
15,
47,
57,
61
FFT:
Exercises
7.4 / 1, 3, 59
-
Lecture 19 (Wed, Oct 7)
Integration of rational functions by partial fractions:
partial fraction expansion when the denominator is a product
of factors that do not have real zeros; examples;
a digression: computing integral of sec(x) in four different ways:
(1) as on page 499 of the book;
(2) by multiplying the numerator and denominator by cos(x),
making the substitution u=sin(x),
and using the formula for the derivative of inverse hyperbolic tangent;
(3) by multiplying the numerator and denominator by cos(x),
making the substitution u=sin(x),
and using the method of partial fractions to represent 1/(1−x2);
(4) by using the Weierstrass substitution t=tan(x/2)
from Exercise 7.4/59
[pages 513-516 of Sec. 7.4]
Homework:
Exercises
7.4 /
6,
21,
23,
39, 52
FFT:
Exercises
7.4 /
31,
56, 65, 66
The complete Homework 6
(problems given on October 2, 5, 7) is due on October 14 (Wednesday).
-
Lecture 20 (Mon, Oct 12):
A digression: functions whose antiderivative cannot be expressed
in terms of elementary function; a detailed study of the function
F(x) defined as a integral from t=0 to t=x
of exp(−t2), a proof that limit of F(x)
as x→∞ is finite by giving an upper bound
on exp(−t2) by e−t
for t≥1, and computing the integral of e−t explicitly.
Strategy for integration:
a discussion of the different methods for integration studied so far;
Examples 1-5; Exercises 7.5/14,
7.5/17,
7.5/32, 7.5/52, 7.5/66
[Sec. 7.5]
Homework:
Exercises
7.5 /
22,
23,
31,
45,
49,
71
FFT:
Exercises
7.5 /
7,
41,
57,
63
-
Lecture 21 (Wed, Oct 14):
Strategy for integration (cont.):
expressing integrals in terms of
special functions
(elliptic integrals,
Bessel functions,
Hankel functions,
exponential integral,
gamma function,
spherical harmonics,
and many others);
solving integrals of 1/(x3+1), 1/(x4+1)
[Sec. 7.5]
Homework:
Exercises
7.5 /
64, 66, 70, 84;
additional problem 1 (the additional problems
are not FFT problems, and should be turned in with the regular homework)
FFT:
Exercises
7.5 / 83;
Ch 7 Review (pages 554-556) / 34, 35, 37
-
Lecture 22 (Fri, Oct 16):
Approximate integration:
construction of the definite integral as a limit of Riemann sums;
left and right Riemann sums;
Midpoint Rule; Trapezoidal Rule;
error bounds for the Midpoint and the Trapezoidal Rules
- for both methods the errors behave like n−2,
i.e., like like h2 (where h=(b−a)/n
is the spacing between two consecutive points xi)
[pages 530-535 of Sec. 7.7]
Homework:
Exercises
7.7 /
2 (give detailed explanations!),
47,
49;
Ch 7 Review (pages 554-556) / 2, 7, 12;
additional problem 2 (the additional problems
are not FFT problems, and should be turned in with the regular homework)
FFT:
Exercises
7.7 /
1,
3,
48;
Ch 7 Review (pages 554-556) / 1, 24, 55
The complete Homework 7
(problems given on October 12, 14, 16) is due on October 21 (Wednesday).
-
Lecture 23 (Mon, Oct 19):
Approximate integration (cont.):
interpolation - given a set of points
(x0,y0),
(x1,y1),
...,
(xn,yn)
with x0<x1<...<xn,
constructing a function p(x) ("interpolant")
from a class of "nice" functions (e.g., polynomials)
that satisfies p(xj)=yj
for j=0,1,...,n;
linear interpolation - constructing the unique linear function
Ax+B
through two points,
(x0,y0) and
(x1,y1),
with x0<x1;
quadratic interpolation - constructing the unique linear function
Ax2+Bx+C
through three points,
(x0,y0),
(x1,y1),
and
(x2,y2),
with x0<x1<x2;
Simpson's formula for aproximate integration
- approximating the integrand locally by quadratic polynomials
and integrating the resulting quadratic function;
detailed derivation of Simpson's formula;
error bound of Simpson's formula - behaves like n−4,
i.e., like h4 (where h=(b−a)/n
is the spacing between two consecutive points xi)
[pages 535-539 of Sec. 7.7]
Homework:
Exercises
Ch 7 Review (pages 554-556) / 59;
Ch 7 Problems Plus (pages 557-559) / 9;
additional problem 1 (the additional problems
are not FFT problems, and should be turned in with the regular homework)
FFT:
Ch 7 Concept Check (page 553) / 1-5;
Ch 7 True-False Quiz (page 554) / 1-5, 8, 9
-
Lecture 24 (Wed, Oct 21):
Improper integrals:
two types of possible problems with integrals - infinite integration interval
and discontinuous integrand;
definition of an improper integral of type 1 as a limit of an integral over
a finite interval [a,b] as
a→−∞ and/or b→∞;
improper integrals of type 2 (whose integrand has an infinite discontinuity
at an endpoint or at an internal point of the integration interval);
definition of an improper integral of type 2 over [a,b]
as an one-sided limit of an integral over a subinterval of [a,b]
when the discontinuity of the integrand occurs at an endpoint;l
definition of an improper integral of type 2 as the sum of two one-sided limits
when the discontinuity of the integrand occurs inside (a,b);
dealing with "doubly" improper integrals, i.e., integrals over infinite regions
and with an integrand that has an infinite discontinuity
- break the integral into a sum of an integral of type 1 and an integral of type 2
(by definition, the improper integral converges only if both integrals converge);
examples
[pages 543-549 of Sec. 7.8]
Homework:
Exercises
7.8 / 32, 55, 76, 78;
Ch 7 Review (pages 554-556) / 46
FFT:
Exercises
7.8 /
1,
2,
13,
29,
31
-
Lecture 25 (Fri, Oct 23):
Improper integrals (cont.):
a Comparison Test for improper integrals; examples;
please read the additional examples
[pages 549, 550 of Sec. 7.8]
Homework:
Exercises
7.8 / 52, 58, 63,
69,
71;
Ch 7 Review (pages 554-556) / 71(a), 79
FFT:
Exercises
7.8 /
21,
49,
57,
75, 79;
Ch 7 Review (pages 554-556) / 49, 71(b);
Ch 7 Concept Check (page 553) / 6-8;
Ch 7 True-False Quiz (page 554) / 6, 7, 10-14
The complete Homework 8
(problems given on October 19, 21, 23) is due on October 28 (Wednesday).
-
Lecture 26 (Mon, Oct 26):
Improper integrals (cont.):
a counterexample to the statement
"If ƒ is continuous on [1,∞) and ƒ(x)→0 as x→∞,
then the integral of ƒ(x) over x∈[1,∞) converges.";
a counterexample to the statement
"If ƒ is continuous and
the integral of ƒ(x) over x∈[1,∞) converges,
then ƒ(x)→0 as x→∞.";
a challenge: find a counterexample to the statement
"If ƒ is continuous on [1,∞) and
the integral of ƒ(x) over x∈[1,∞) converges,
then ƒ(x) is bounded on [1,∞)."
Sequences:
sequences; recursively defined sequences
(example: Fibonacci sequence);
limit of a sequence;
convergent and divergent sequences;
examples of finding limits directly from the definition of a limit;
please read the additional examples
[pages 714-716 of Sec. 11.1]
Homework:
Exercises
11.1 / 8, 12, 14,
the following problems should be solved by using only the definition of convergence:
24,
25,
42,
43
FFT:
Exercises
11.1 / 11,
17
-
Lecture 27 (Wed, Oct 28):
Sequences (cont.):
using your knowledge of limits of functions in order to find limits of sequences:
Theorem 3 (sometimes used together with the l'Hospital's rule),
Theorem 7: limƒ(an)=ƒ(L)
if the function ƒ(x) is continuous at x=L;
simpler ways to find limits of sequences: limit laws;
Squeeze Theorem; using the Squeeze Theorem to show that
if lim|an|=0,
then lim(an)=0;
remark that lim|an|=5,
does not imply that lim(an)=5;
using simple limits: lim(1/nr)=0 if r>0,
lim(cn)=0 if |c|<1,
lim(cn)=∞ if |c|>1;
definition of an infinite limit, lim(an)=∞
[pages 717-720 of Sec. 11.1]
Homework:
Exercises
11.1 /
26 (use directly Definition 5),
28 (see Example 11), 30 (use Theorem 7 and the continuity
of ƒ(x)=x1/2 at x=0),
36 (use Theorem 7, specifying for which function and at which point),
38, 48 (use Theorem 6), 49
FFT:
Exercise
11.1 /
53
-
Discussion 27a (Thu, Oct 29):
Exam 2
[on Mathematical Induction (App. E) and Sec. 7.1-7.5, 7.7, 7.8,
covered in Lectures 13-25]
-
Lecture 28 (Fri, Oct 30):
Sequences (cont.):
convergence of sequences - examples
[pages 714-720 of Sec. 11.1]
-
Lecture 29 (Mon, Nov 2):
Sequences (cont.):
increasing, decreasing, and monotone sequences; examples;
bounded above, bounded below, and bounded sequences; examples;
Monotonic Sequence Theorem;
using the Monotonic Sequence Theorem to prove existence of a limit of a recursively
defined sequence, and then finding the limit of the sequence
(Example 14)
[pages 720-723 of Sec. 11.1]
Homework:
Exercises
11.1 / 70, 74, 76, 78, 80,
81
FFT:
Exercise
11.1 /
71,
73,
79, 91
-
Lecture 30 (Wed, Nov 4):
Sequences (cont.):
computing the resistance of an infinite chain of resistors
as a limit of a recursively defined sequence;
the base e of the natural logarithms as a limit of the sequence
(1+1/n)n.
Series:
series; partial sums; convergent and divergent series;
geometris series (Examples 2-6); telescoping series (Example 7);
if Σan converges,
then liman=0 (Theorem 6);
the converse is not true (example: harmonic series, to be discussed in Lecture 31);
Test for Divergence by proving that liman≠0;
facts about Σcan,
Σ(an+bn),
Σ(an−bn)
for convergent series
Σan and
Σbn and a constant c (Theorem 8)
[Sec. 11.2, skip Example 8]
Homework:
Exercises
11.2 / 4, 18, 22, 24, 30, 33, 36, 44, 45, 52
FFT:
Exercises
11.2 /
15,
16,
23,
39,
43,
51,
79,
additional FFT problem
The complete Homework 9
(problems given on October 26, 28, November 2, 4) is due on November 9 (Monday).
-
Lecture 31 (Fri, Nov 6):
Series:
proof of Theorem 6;
harmonic series as an example of a divergent series
whose terms an converge to zero (Example 8)
[pages 732, 733 of Sec. 11.2]
The Integral Test and estimates of sums:
the Integral Test (with proof);
using the Integral Test to prove that the harmonic series diverges
[pages 738-741, 744 of Sec. 11.3]
Reading assignment:
p-test; examples
[page 741 of Sec. 11.3]
Thinking assignment:
thinking geometrically as the terms an
as an area of a rectangle with width 1 and height an,
give an upper and lower bounds on the error,
|s−sn|=|sn+1+sn+2+…|,
in approximating the sum s of a convergent series with
positive decreasing terms (an>an+1>0 ∀n∈N)
by its nth partial sum sn,
in terms of an integral from an appropriately chosen function.
Homework:
Exercises
11.2 / 62, 63,
73,
86;
11.3 / 8, 13, 28, 29, 34,
45, 46 (in Exercise 45 use that
blnn=e(lnb)(lnn)=nlnb
and apply the p-test; in Exercise 46 don't split the series into two series)
FFT:
Exercises
11.2 /
57,
59,
67,
85;
11.3 /
7,
11,
21
-
Lecture 32 (Mon, Nov 9):
The Integral Test and estimates of sums (cont.):
remainder estimate for the Integral Test; estimating the sum of a series; example;
remarks on using the Integral Test when the terms an do not decrease,
but only decrease eventually (i.e., for all n greater than some large N);
an example with Σnn1000e−n
[pages 742, 743 of Sec. 11.3]
The Comparison Test:
the Comparison Test; the Limit Comparison Test; estimating sums; examples
[Sec. 11.4]
Homework:
Exercises
11.3 / 36, 44;
11.4 /
11, 13, 27, 29,
41,
additional problem 1
FFT:
Exercises
11.3 /
37,
41;
11.4 /
1,
2,
5,
7,
17,
31,
40, 42
-
Lecture 33 (Wed, Nov 11):
Alternating series:
definition of alternating series; the Alternating Series Test (with a sketch of proof);
Alternating Series Estimation Theorem; examples
[Sec. 11.5]
Homework:
Exercises
11.5 / 4, 10, 12, 16, 18, 20, 25, 30, 36 (in Exercise 36 you may use the results
of Exercise 11.3/44),
additional problem 2
FFT:
Exercises
11.5 /
3,
7,
11,
17,
23,
32,
additional FFT problem
The complete Homework 10
(problems given on November 6, 9, 11) is due on November 16 (Monday).
-
Lecture 34 (Fri, Nov 13):
Absolute convergence and the Ratio and Root Tests:
absolutely convergent series; conditionally convergent series;
absolute convergence implies convergence
(but convergence does not imply absolute convergence!);
the Ratio Test (with the proof that
lim|an+1/an|=L<1
implies absolute converence of
Σnan);
examples of using the Ratio Test;
examples illustrating the fact that
lim|an+1/an|=1
can happen for absolutely convergent, conditionally convergent,
or non-convergent series
[pages 756-760 of Sec. 11.6]
Thinking assignment:
prove the part of the Ratio Test that
lim|an+1/an|=L>1
implies that Σnan
does not converge.
Homework:
Exercises
11.6 / 5, 6, 8, 10,
13,
19,
20, 24,
31,
33.
Remarks and hints: You have to check for convergence and for absolute convergence!
Do not forget about the Comparison Test, the Limit Comparison Test, the p-test,
the Alternating Series Test,
the Ratio and Root Tests
FFT:
Exercises
11.6 /
3,
35,
37
-
Lecture 35 (Mon, Nov 16):
Absolute convergence and the ratio and root tests (cont.):
the Root Test;
rearrangement of the "alternating harmonic series"
1−1/2+1/3−1/4+1/5−1/6+1/7-1/8+...
to obtain a series whose sum is half of the sum of the original series;
Riemann Rearrangement Theorem:
if an infinite series is conditionally convergent (i.e., convergent but not
absolutely convergent), then its terms can be rearranged (i.e., taken
in a different order) so that the new series converges to any given value, or diverges
[page 761 of Sec. 11.6]
Reading assignment (mandatory):
strategy for testing series [Sec. 11.7]
Homework:
Exercises
11.6 / 14, 15, 22, 29, 32, 34, 38
FFT:
Exercises
11.6 /
21,
23, 42.
Hints: Exercise 42(a): assume that Σn2an converges,
which implies something about lim(n2an),
which in turn can be used to show that there exists an N such that
0≤n2|an|<1 for all n≥N,
i.e.,
0≤|an|<1/n2 for all n≥N,
which would imply something about the convergence of the series Σan
that would lead to a contradiction;
Exercise 42(b): think about the series
Σ(−1)n(n ln n)−1,
Σ(−1)n−1/n,
Σ(−1)n−1/n1/2
-
Lecture 36 (Wed, Nov 18):
Power series:
definition of a power series;
examples of power series that converge at one point only,
of power series that converge on an interval (of the form (a,b),
(a,b], [a,b), or [a,b]),
and of power series that converge on the whole real line;
Theorem that the above three possibilities are the only three things
that can happen (Theorem 3);
interval of convergence and radius of convergence of a power series;
examples of finding the interval of convergence of a powers series
[Sec. 11.8]
Homework:
Exercises
11.7 / 6, 14, 23, 24, 33;
11.8 / 10, 17,
37
(in the exercises from Sec 11.8 investigate each endpoint of the interval of convergence).
Hints for the exercises from Sec. 11.7 (please try to solve the problems before you look at the hints):
6 - Limit Comparison Test or Integral Test,
14 - Comparison with a geometric series,
23 - Limit Comparison Test with a well-known series,
24 - Test for Divergence,
33 - Root Test.
FFT:
Exercises
11.7 / 2, 4, 8, 10, 20, 22, 36;
11.8 / 4,
24,
26;
Ch 11 Concept Check (page 802) / 1-8.
Hints for the problems from Sec. 11.7 (please try to solve the problems before you look at the hints):
2 - Root Test,
4 - Alternating Series Test,
8 - Ratio Test or Test for Divergence,
10 - Integral Test,
20 - Comparison with a p-series,
22 - Test for Divergence,
36 - (ln n)ln n=(eln ln n)ln n=(eln n)ln ln n=nln ln n>n2
for all n greater than some N because ln ln n→∞.
The complete Homework 11
(problems given on November 13, 16, 18) is due on November 23 (Monday).
-
Lecture 37 (Fri, Nov 20):
A digression:
Martin Gardner,
the best friend Mathematics ever had.
Representations of functions as power series:
examples of using the formula for the sum of a geometric series to derive
representations of functions as power series;
Theorem on term-by-term differentiation and integration of power series;
more examples using differentiation and integration of power series:
power series representations of ln(1+x) (Example 6)
[pages 770-774 of Sec. 11.9]
Reading assignment:
power series expansion of arctan(x) (Example 7)
[page 774 of Sec. 11.9]
Homework:
Exercises
11.9 /
8,
14 (see Example 6),
17 (see Example 5, first expand 1/(1+4x)2 by differentiating 1/(1+4x)),
23,
25,
additional problem
FFT:
Exercises
11.9 /
5,
13(a,b),
15,
37,
39;
Ch 11 True-False Quiz (page 802, 803) / 1-9, 11, 12, 14-20
-
Lecture 38 (Mon, Nov 23):
Representations of functions as power series (cont.):
using a power series expansion of ƒ(x) to compute approximately
the value of a definite integral of ƒ(x) (Example 8)
[page 774, 775 of Sec. 11.9]
Taylor and Maclaurin series:
Taylor series of a function ƒ at a point a (or "about a"
or "centered at a"); Maclaurin series;
nth degree Taylor polynomial Tn(x)
and remainder Rn(x), such that
ƒ(x)=Tn(x)+Rn(x);
Theorem about the converence of the remainder Rn(x)
to 0 as n→∞; Taylor's inequality - an upper bound on the size
of the remainder:
|Rn(x)|≤M|x−a|n+1/(n+1)!,
where M is the maximum value of |ƒ(n+1)(ξ)| for ξ between a and x;
examples
[pages 777-782 of Sec. 11.10]
Homework:
Exercises
11.9 /
32, 40, 42 (use that 1+x3=(1+x)(1−x+x2));
11.10 /
8, 11, 14, 44 (use the Alternating Series Estimation Theorem to make sure that the truncation
error does not exceed 10−5)
FFT:
Exercises
11.9 / 36, 38 (illustrating the dangers of differentiating series of functions);
11.10 / 1, 2, 9,
15,
57,
63;
Ch 11 Concept Check (page 802) / 9-11;
Ch 11 True-False Quiz (page 802, 803) / 10, 13
-
Lecture 39 (Mon, Nov 30):
Taylor and Maclaurin series (cont.):
Taylor and Maclaurin series for ex,
sin(x), cos(x), sinh(x), cosh(x);
Binomial Series; using series to evaluate indefinite and definite integrals,
using series to evaluate limits
[pages 782-787 of Sec. 11.10]
Reading assignment:
proof
of Taylor's Theorem.
Homework:
Exercises
11.10 /
35,
48, 56, 64, 68, 70
FFT:
Exercises
11.10 /
27,
55, 65, 72;
Ch 11 Concept Check (page 802) / 12;
Ch 11 True-False Quiz (page 802, 803) / 21, 22
The complete Homework 12
(problems given on November 20, 23, 30) is due on December 2 (Wednesday).
-
Lecture 40 (Wed, Dec 2):
Taylor and Maclaurin series (cont.):
multiplication of Taylor series
[pages 787-788 of Sec. 11.10]
Applications of Taylor polynomials:
approximating functions by polynomials;
practical advice - use periodicity of functions
(e.g., when trying to compute sin(5763.78)),
try to make the increment small (e.g.,
when trying to compute approximately
1.461/2, use the Binomial Series to expand
(1.44+0.02)1/2=1.2(1+0.02/1.44)1/2≈1.2(1+(1/2)(0.02/1.44)+...),
instead of using the expansion
1.461/2=(1+0.46)1/2=1+(1/2)0.46+...);
applications to physics - Einstein formula for the energy
of a body moving with velocity v:
E(v)=m0c2(1−v2/c2)−1/2≈m0c2+m0v2/2+...
(where m0v2/2 is the kinetic energy in classical mechanics)
[pages 792-796 of Sec. 11.11]
Homework:
Exercises
11.11 /
18(a,b),
33,
35(a,b), 36, 38(a);
Problems Plus (pages 805-808) / 20
FFT:
Exercises
11.10 /
53,
59
-
Discussion 40a (Thu, Dec 3):
Exam 3
[on Sec. 11.1-11.10 covered in Lectures 26-39]
-
Lecture 41 (Fri, Dec 4):
Applications of Taylor polynomials (cont.):
discussion on some problems from Exam 3.
A digression on some famous theorems:
Fermat's Theorem
proved by
Andrew Wiles in 1999;
Four Color Theorem
proved by
Kenneth Appel
and
Wolfgang Haken
in 1976 (using a computer).
Homework:
Exercises
Ch 11 Review (pages 803-804) / 29, 34, 38, 39, 40, 62.
Hints (please try to solve the problems before you look at the hints):
29 - telescoping sum,
34 - geometric series,
38 - Ratio Test,
39 - Limit Comparison Test,
62 - use the Taylor series for ez.
FFT:
Exercises
Ch 11 Review (pages 803-804) / 27, 31, 32, 47, 49.
Hints: 27 - geometric series, 31 - use the Taylor series for ez,
32 - geometric series, 47 - geometric series, 49 - geometric series.
-
Lecture 42 (Mon, Dec 7):
Arc length:
review of the definition of a definite integral (over a finite interval [a,b])
as a limit of Riemann sums approximating the area under the graph of a function
by a sum of areas of thin rectangles;
computing the volume of a solid by slicing it by a family of parallel planes
and approximating the true volume by a sum of the volumes of the thin slices
(each of which is approximately cylindrical, with area of the base ≈Ai,
height Δzi, and volume
≈AiΔzi);
volume of a solid of revolution (in which
Ai≈πri2);
computing the arc length of the graph of a curve defined by
y=ƒ(x), x∈[a,b] by approximating it by a
piecewise linear function ("broken straight line") and using the Pythagorean Theorem
to approximate the length of the arc between the points
Pi−1=(xi−1,ƒ(xi−1))
and
Pi=(xi,ƒ(xi))
by the length of the segment of straight line
connecting Pi−1 and Pi
[pages 288-289 of Sec. 4.1, 296-297 of Sec. 4.2, 352-355 of Sec. 5.2, 562-565 of Sec. 8.1]
Homework:
Exercise
8.1 /
11;
additional problems
-
Lecture 43 (Wed, Dec 9):
Arc length (cont.):
a digression on what a "natural geometric object" means;
the arc length function as a natural parameter
[pages 565-567 of Sec. 8.1]
Volume of an n-dimensional ball:
definition of the ball Bn(R)
of radius R in Rn,
and its volume Vn(R)
and its surface area An(R);
particular cases for n equal to 1, 2, 3;
proof that An(R)=dVn(R)/dR.
Homework:
No homework problems are assigned.
FFT:
Exercise
8.1 /
33
The complete Homework 13
(problems given on December 2, 4, 7, 9) is due on December 11 (Friday).
-
Discussion 43a (Thu, Dec 10):
Volume of an n-dimensional ball (cont.):
recurrence relations between the integral
In
of cosn(x) over [-π/2,π/2]
and derivation of the expressions
I2k=π(2k−1)!!/(2k)!!,
I2k+1=2(2k)!!/(2k+1)!!;
Gamma function: definition, derivation of the property
Γ(x)=(x−1)Γ(x−1),
and using it to obtain that
Γ(n/2+1)=πn/2/In.
-
Lecture 44 (Fri, Dec 11):
Volume of an n-dimensional ball (cont.):
derivation of the volume of B3(R)
by integrating over x3 in [−R,R]
of the "volume" of the cross-section of B3(R)
with the plane {x3=a} (for a∈[−R,R]),
which is a 2-dimensional "ball" of radius (R2−x32)1/2,
and of "volume"
V2((R2−x32)1/2)=π[(R2−x32)1/2]2.
Why is sometimes the interval of convergence of the Taylor series of "nice" function smaller than the domain of the function?:
complex numbers, complex plane C,
singularities of functions in the complex plane,
the Taylor series of functions of complex variables converge in a disk in C
(which can degenerate to a point or the whole C);
example - the function 1/(1+x2) is defined on all of R,
but its Taylor series converges only on (−1,1) - the reason for this is are the singularities
of the function 1/(1+z2) (where z∈C)
at i and −i; same phenomenon happens for the function arctan
because arctan'(x)=1/(1+x2).
Food for thought:
Write Vn(R)=cnRn
and express Vn(R)
as an integral over xn in [−R,R]
of the volume
Vn−1((R2−xn2)1/2)
which is equal to
cn−1[(R2−xn2)1/2]n−1,
use the substitution
xn=Rcos(θ) to express
the integral in terms of the integrals Ik computed in Discussion 43a
and in terms of the Gamma function;
the result is
Vn(R)=πn/2Rn/Γ(n/2+1);
use the relation between An(R) and
Vn(R) to compute the surface area of the n-dimensional ball of radius R.
Fun reading:
How the General Assembly of the State of Inidiana tried to establish mathematical truth by legislative fiat by
redefining the value of π.
-
Final Exam
[cumulative] -
Friday, December 18, 8:00-10:00 a.m., 809 PHSC
Good to know: